where \(G \in SE(3)\)  is an over-parameterization that can cause problems in the unconstrained continuous optimization. To this end, we use the Lie-algebra representation \(\xi \in \mathfrak {se}(3)\)  for the minimal parameterization of \(G\in \text{SE}(3)\) . The exponential map \(G = \text{exp}_{\mathfrak {se}(3)}(\xi )\)  converts \(\mathfrak {se}(3) \mapsto \text{SE}(3)\) , while the log map
\(\xi = \text{log}_{SE(3)}(G)\)  converts \(\text{SE}(3) \mapsto \mathfrak {se}(3)\) . Similar to [1]}, we define the \(\mathfrak {se}(3)\)  concatenation operator \(\circ : \mathfrak {se}(3) \times \mathfrak {se}(3) \mapsto \mathfrak {se}(3)\)  as:
\(\small \xi _{ki} = \xi _{kj} \circ \xi _{ji} = \text{log}_{\text{SE}(3)}\big ( \text{exp}_{\mathfrak {se}(3)}(\xi _{kj}) \cdot \text{exp}_{\mathfrak {se}(3)}(\xi _{ji}) \big ),\) 
